If you work full time, you should get at least 2 weeks of vacation time every year, so you could use that time for video games & for family. I don't get why you would need vacation time for the latter though, besides sudden illness or death, but you shouldn't need vacation time for that either.
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Vacation time, to play a video game? Really?
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It's meant to take time away from work to do whatever you want, so yes, if they want to.